TY - JOUR
T1 - New insights on free energies and Saint-Venant's principle in viscoelasticity
AU - Deseri, L.
AU - Gentili, G.
AU - Golden, J. M.
N1 - Funding Information:
L. Deseri gratefully acknowledges financial support from the grant PIAPP-GA-2013-609758-HOTBRICKS, “Mechanics of refractory materials at high temperature for advanced industrial technologies”, from the EU through the FP7 program. The Center for Nonlinear Analysis at Carnegie Mellon University through the NSF Grant No. DMS-0635983 is also acknowledged. Both the departments of Civil and Environmental Engineering and of Mechanical Engineering at Carnegie Mellon University are gratefully acknowledged for their support and hospitality to L. Deseri during the Spring 2014. The Department of Civil, Environmental and Mechanical Engineering DICAM from the University of Trento is also gratefully acknowledged for the permission granted to L. Deseri to actively be affiliated with Carnegie Mellon and with The Houston Methodist Research Institute and to partially pursue his research there.
PY - 2014/10/1
Y1 - 2014/10/1
N2 - Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll's definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation. In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space-time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space-time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy. Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case. The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.
AB - Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll's definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation. In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space-time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space-time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy. Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case. The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.
KW - Dissipation rate
KW - Free energy
KW - Residual stress decay
KW - Saint Venant principle
KW - Spatial decay
KW - State in viscoelasticity
KW - Viscoelasticity
UR - http://www.scopus.com/inward/record.url?scp=84906279343&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84906279343&partnerID=8YFLogxK
U2 - 10.1016/j.ijsolstr.2014.05.031
DO - 10.1016/j.ijsolstr.2014.05.031
M3 - Article
AN - SCOPUS:84906279343
VL - 51
SP - 3382
EP - 3398
JO - International Journal of Solids and Structures
JF - International Journal of Solids and Structures
SN - 0020-7683
IS - 19-20
ER -